Diophantine Approximation Festschrift for Wolfgang Schmidt

This volume contains 22 research and survey papers on recent developments in the field of diophantine approximation. The first article by Hans Peter Schlickewei is devoted to the scientific work of Wolfgang Schmidt. Further contributions deal with the subspace theorem and its applications to diophantine equations and to the study of linear recurring sequences. The articles are either in the spirit of more classical diophantine analysis or of geometric or combinatorial flavor. In particular, estimates for the number of solutions of diophantine equations as well as results concerning congruences and polynomials are established. Furthermore, the volume contains transcendence results for special functions and contributions to metric diophantine approximation and to discrepancy theory. The articles are based on lectures given at a conference at the Erwin Schr6dinger Institute in Vienna in 2003, in which many leading experts in the field of diophantine approximation participated. The editors are very grateful to the Erwin Schr6dinger Institute and to the FWF (Austrian Science Fund) for the financial support and they express their particular thanks to Springer-Verlag for the excellent cooperation. Robert E Tichy Diophantine Approximation H. E Schlickewei et al. , Editors 9 Springer-Verlag 2008 THE MATHEMATICAL WORK OF WOLFGANG SCHMIDT Hans Peter Schlickewei Mathematik Informatik, und Philipps-Universitiit Hans-Meerwein-Strasse, Marburg, 35032 Marburg, Germany k. Introduction Wolfgang Schmidt's mathematical activities started more than fifty years ago in 1955. In the meantime he has written more than 180 papers - many of them containing spectacular results and breakthroughs in different areas of number theory.

The Mathematical Work of Wolfgang Schmidt.- SchÄffer’s Determinant Argument.- Arithmetic Progressions and Tic-Tac-Toe Games.- Metric Discrepancy Results for Sequences {nkx} and Diophantine Equations.- Mahler’s Classification of Numbers Compared with Koksma’s, II.- Rational Approximations to A q-Analogue of ? and Some Other q-Series.- Orthogonality and Digit Shifts in the Classical Mean Squares Problem in Irregularities of Point Distribution.- Applications of the Subspace Theorem to Certain Diophantine Problems.- A Generalization of the Subspace Theorem With Polynomials of Higher Degree.- On the Diophantine Equation G n (x) = G m (y) with Q (x, y)=0.- A Criterion for Polynomials to Divide Infinitely Many k- Nomials.- Approximants de Padé des q-Polylogarithmes.- The Set of Solutions of Some Equation for Linear Recurrence Sequences.- Counting Algebraic Numbers with Large Height I.- Class Number Conditions for the Diagonal Case of the Equation of Nagell and Ljunggren.- Construction of Approximations to Zeta-Values.- Quelques Aspects Diophantiens des VariéTés Toriques Projectives.- Une Inégalité de ?ojasiewicz Arithmétique.- On the Continued Fraction Expansion of a Class of Numbers.- The Number of Solutions of a Linear Homogeneous Congruence.- A Note on Lyapunov Theory for Brun Algorithm.- Orbit Sums and Modular Vector Invariants.- New Irrationality Results for Dilogarithms of Rational Numbers.